Re terminology:

[Ronnie Brown <ronnie.profbrown@btinternet.com>]

Peter Sellinger writes recently: 

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I think this is a very apt illustration of what happens if a term with
an existing meaning is redefined to mean something else. Henceforth it
is impossible for anybody to use the term (with either meaning)
without first giving a definition.

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I completely agree. My own problem is with term `infinity groupoid' which  is used to describe something which is not even a groupoid, and whose use seems to me to militate against the understanding of what has been achieved with the original and much earlier definition. I once asked Gian-Carl Rota about such change of terminology, in connection with a refereeing job, and he agreed that mathematicians are used to creating confusion in this way. 

There are two easy tendencies: one is to use an old name in a quite different way, and the other is to use a new name for an old idea, so that the  use of the old term looks old fashioned, and a lot of work may be consigned to the dustbin of history, becoming not easy of access  for new students. 

It seems to be an example of these confusions is the way the simplicial singular complex of a space is called an infinity-groupoid, even the `fundamental infinity groupoid', when what seems to be referred to is that it is a Kan complex, i.e. satisfies the Kan extension condition, studied since 1955. The new term sounds like `dressing up' an old idea to look new. My personal objection to this change of terminology (i.e. axe to grind!) is that this distracts from studying the not so simple proofs that strict  higher homotopical structures exist, which mainly are for structured spaces (in particular filtered spaces (Brown/Higgins, Ashley), n-cubes of spaces (Loday), and more recently smooth spaces (Faria Martins/Picken)). The analysis and comparison of these uses should be made. It was certainly a relief to Philip and I that we could do something with filtered spaces which we could not do for the absolute case; the significance of the fact  that these constructions work and lead to specific calculations should be thought about. 

The term `higher dimensional group theory' which was published in a paper  with that title in 1982 was intended to suggest developing higher groupoid theory and its relations to homotopy theory in the spirit of group theory, which meant specific constructions relevant to geometry and calculations, even computer calculations,  of many examples, in which actual numbers arise as a test of and examples of the general theory, and in which some aspects of group theory are sensibly seen as better represented in the higher dimensional theory; and example of this is the nonabelian tensor  product of groups, where group theorists have found lots of pickings. 

I am not sure how these terminological problems will be resolved, and I know the term (\infty,n)-groupoid has been well used recently but the problem of relation to the older ideas, which have had a certain success, should be recognised. 

Ronnie Brown








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